In what ratio is the line segment joining $(-3, -1)$ and $(-8, -9)$ divided at the point $(-5, −\frac{21}{5})$?

Given:

The line segment joining the points $(-3, -1)$ and $(-8, -9)$ is divided at the point $(-5, −\frac{21}{5}).

To do:

We have to find the ratio of the division.

Solution:

Let the point \( \left(-5, \frac{-21}{5}\right) \) divides the line segment joining the points \( (-3,-1) \) and \( (-8,-9) \) in the ratio of \( m: n \).

Using section formula, we have,

\( (x, y)=(\frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n}) \)

Therefore,

\( x=\frac{mx_{2}+nx_{1}}{m+n} \)
\( \Rightarrow -5=\frac{m(-8)+n(-3)}{m+n} \)

\( \Rightarrow -5=\frac{-8m-3n}{m+n} \)
\( \Rightarrow -5m-5n=-8m-3n \)
\( \Rightarrow -5 m+8 m=-3 n+5 n \)

\( \Rightarrow 3 m=2 n \)
\( \Rightarrow \frac{m}{n}=\frac{2}{3} \)

\( \Rightarrow m:n=2:3 \)

The required ratio is $2:3$.

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Updated on: 10-Oct-2022

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